The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 391

EXAMPLE 6.9 Interpolation Functions for Rectangular Elements
For the normalized rectangular element, with centroidal coordinates

, of Fig. 6.35, we will
demonstrate the use of Lagrange and Hermitian polynomials in developing interpolation
functions.
For four nodes, form Eq. (6.65b),

ξη

] v 1

N 1 (η)
N 2

v 4

u (ξ , η) = [ N 1 (ξ ) N 2 (ξ )

v v (η)

2 3

= N 1 (ξ ) N 1 (η)v 1 + N 2 (ξ ) N 1 (η)v 2 + N 1 (ξ ) N 2 (η)v 4 + N 2 (ξ ) N 2 (η)v 3

= N 1 (ξ , η)v 1 + N 2 (ξ , η)v 2 + N 3 (ξ , η)v 3 + N 4 (ξ , η)v 4 (1)

From Eq. (6.64)

N 1 (ξ ) = ξ − ξ

ξ − 1 1

2 (

2

2 =

1 =

1 − ξ)

ξ − ξ − 1 −

1

N 2 (ξ ) = ξ − ξ 1

ξ + 1 1

2 ( ξ 2 − ξ 1 = ) =

1 + ξ)

1 − ( − 1

(2)

(η) = η − η

η −

1 1

2 (

4

N 1

4 =

1 =

1 − η)

η − η − −

1

1

N 2 (η) = η − η 1
η

η + 1 1

2 (

1 =

) =

1 + η)

− η 1 − ( − 1

4

so that

1
4 (

1
4 (

N 1 (ξ , η) = 1 − ξ)( 1 − η) N 2 (ξ , η) = 1 + ξ)( 1 − η)

(3)

1
4 (

1
4 (

(ξ η) = + ξ)( + η) (ξ η) = − ξ)( + η)

N 3 , 1 1 N 4 , 1 1

If derivatives, such as rotations, of u are to be included at the nodes or if greater than C 0
continuity is desired at the nodes, then use Hermitian interpolation. At each node, introduce
the four DOF

2 u
∂ξ∂η

v ξ = ∂

u
∂ξ

v η = ∂

u
∂η

v ξη = ∂ v = u, , , (4)

and assume that Eq. (6.89) applies, with N i ξ =

(ξ )

and N i η =

(η)

. The interpola-
tion functions are obtained by utilizing third degree Hermitian polynomials of Chapter 4,
Eq. (4.47b) for N i ξ

N i N i

and N i η . For example, for the term corresponding to the v

1

2 3 2 3

N 1 (ξ , η) = N 1 ξ N 1 η = ( 1 − 3 ξ + 2 ξ )( 1 − 3 η − 2 η ) (5)

6.5.8 Stiffness Matrices and Loading Vectors For Triangular and Tetrahedral Elements

This section contains the formulation of the stiffness matrices and the loading vectors for
triangular and tetrahedral elements using the natural coordinates of Eqs. (6.75) and (6.82).

Triangular Element
Consider a plate under plane stress conditions. Divide the plate into three-node triangular
elements of the sort shown in Fig. 6.30. Let u x ,u y be the displacements in the global x and
y directions. Traditionally, this notation is used, although in the previous chapter X and Y